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Fibonacci Sequence Calculator

Compute the nth Fibonacci number with fast BigInt arithmetic, generate Fibonacci sequences and sums, test if a number is Fibonacci and explore connections with Binet’s formula, the golden ratio and custom recurrences.

Nth Fibonacci (BigInt) Sequence Generator & Sums Fibonacci Number Test Golden Ratio & Recurrences

Advanced Fibonacci Sequence Calculator

This Fibonacci Sequence Calculator goes beyond just computing a few terms. It uses a fast doubling algorithm with high-precision integers to find the nth Fibonacci number, generate sequences, compute sums, test whether a given integer is Fibonacci and illustrate formulas involving the golden ratio and linear recurrences.

By default, this calculator uses the standard indexing convention \[ F_0 = 0,\quad F_1 = 1,\quad F_n = F_{n-1} + F_{n-2}\ \text{for}\ n \ge 2. \] A fast doubling algorithm computes \(F_n\) exactly as an integer for reasonably large \(n\).

Generate the Fibonacci sequence from \(F_0\) up to \(F_n\) and see the values in both a table and a compact list. For performance and readability, the maximum index for the table is capped.

The sum of the first \(n + 1\) Fibonacci numbers satisfies the identity \[ \sum_{k=0}^{n} F_k = F_{n+2} - 1. \] This tab uses the fast doubling algorithm to compute \(F_{n+2}\) and then returns the exact sum.

A positive integer \(N\) is Fibonacci if and only if one of \[ 5N^2 + 4,\quad 5N^2 - 4 \] is a perfect square. This tab uses that property with high-precision integer arithmetic to test whether a given integer is in the Fibonacci sequence.

Binet’s closed-form expression for the Fibonacci sequence is \[ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}, \] where \[ \varphi = \frac{1 + \sqrt{5}}{2},\quad \psi = \frac{1 - \sqrt{5}}{2}. \] This tab compares exact values with the approximation obtained from the closed-form expression.

As \(n\) grows, the ratio of successive Fibonacci numbers approaches the golden ratio \[ \varphi = \frac{1 + \sqrt{5}}{2}. \] This tab explores the ratio \(F_{n+1} / F_n\) and compares it with \(\varphi\) numerically.

Many sequences share the same recurrence as Fibonacci but start from different initial values. This tab lets you define a sequence \[ a_0 = A,\quad a_1 = B,\quad a_n = a_{n-1} + a_{n-2} \] and then computes \(a_n\) along with a small table of terms.

Fibonacci Sequence Calculator – Nth Term, Sums and Golden Ratio

The Fibonacci Sequence Calculator on MyTimeCalculator provides an advanced toolkit for working with one of the most famous integer sequences in mathematics. It uses high-precision arithmetic to compute the nth Fibonacci number, generate tables of values, evaluate exact sums, test whether an integer is Fibonacci and explore theoretical connections with closed-form formulas, the golden ratio and linear recurrences.

The Fibonacci sequence appears in number theory, computer algorithms, combinatorics, geometry, finance and models of growth. Many real-world growth processes are only approximately Fibonacci, but the underlying patterns are often similar to those captured by this simple recurrence.

1. Definition of the Fibonacci Sequence

The most common definition of the Fibonacci sequence uses the two initial conditions

\[ F_0 = 0,\quad F_1 = 1, \]

followed by the recurrence relation

\[ F_n = F_{n-1} + F_{n-2}\quad\text{for}\ n \ge 2. \]

The first few terms are:

\[ 0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\dots \]

Each term is the sum of the two preceding terms. This simple rule leads to a wide range of identities and surprising connections with geometry, continued fractions and algebra.

2. Fast Doubling Algorithm for \(F_n\)

Directly applying the recurrence relation from \(F_0\) up to \(F_n\) takes \(O(n)\) steps. The calculator uses a fast doubling method that reduces the complexity to \(O(\log n)\) by exploiting identities such as

\[ F_{2k} = F_k \bigl(2F_{k+1} - F_k\bigr),\quad F_{2k+1} = F_{k+1}^2 + F_k^2. \]

By repeatedly splitting the index in half, the algorithm can compute very large Fibonacci numbers efficiently. Combined with high-precision integers, this allows the Nth Fibonacci tab to return exact values for large indices where ordinary calculators would overflow.

3. Sums of Fibonacci Numbers

Fibonacci sums obey a particularly neat identity:

\[ \sum_{k=0}^{n} F_k = F_{n+2} - 1. \]

This means that once you can compute \(F_{n+2}\) quickly, the sum of the first \(n+1\) terms is available immediately. The Sum of First n Terms tab in the calculator uses exactly this identity, combining it with the fast doubling algorithm for \(F_{n+2}\).

4. Testing Whether a Number Is Fibonacci

There is an elegant characterization of Fibonacci numbers in terms of perfect squares. A non-negative integer \(N\) is a Fibonacci number if and only if one of the two integers

\[ 5N^2 + 4,\quad 5N^2 - 4 \]

is a perfect square. The Fibonacci Number Test tab implements this criterion using high-precision arithmetic and an integer square-root routine. If either expression is a perfect square, the input is classified as a Fibonacci number; otherwise, it is not.

5. Binet’s Closed-Form Expression

Although the Fibonacci sequence is defined by a simple recurrence, it also admits a closed-form expression involving the golden ratio. Let

\[ \varphi = \frac{1 + \sqrt{5}}{2},\quad \psi = \frac{1 - \sqrt{5}}{2}. \]

Then for all integers \(n \ge 0\),

\[ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}. \]

Because \(|\psi| < 1\), the term \(\psi^n\) rapidly becomes very small in magnitude as \(n\) grows, so \(\varphi^n / \sqrt{5}\) is already extremely close to \(F_n\) for moderate \(n\). The closed-form tab in the calculator compares the exact integer value with the value obtained from this expression and reports the difference.

6. Golden Ratio Limits

One of the most famous properties of the Fibonacci sequence is that the ratio of consecutive terms tends to the golden ratio as \(n\) increases:

\[ \lim_{n \to \infty} \frac{F_{n+1}}{F_n} = \varphi. \]

The Golden Ratio tab evaluates the ratio \(F_{n+1} / F_n\) for your chosen index \(n\) and compares it to a numerical approximation of \(\varphi\). As \(n\) grows, the ratio converges rapidly, illustrating the connection between linear recurrences and characteristic roots.

7. Custom Recurrences of Fibonacci Type

Many sequences in applications follow the same recurrence relation as the Fibonacci sequence but start from different initial values. If we choose

\[ a_0 = A,\quad a_1 = B,\quad a_n = a_{n-1} + a_{n-2}, \]

the resulting sequence is a linear combination of Fibonacci numbers:

\[ a_n = A\,F_{n-1} + B\,F_n\quad\text{for}\ n \ge 1. \]

The Custom Recurrence tab calculates \(a_n\) directly from the recurrence and produce a small table of terms. This is useful for modelling processes that start from custom initial conditions but otherwise follow the same additive pattern.

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Fibonacci Sequence Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about the Fibonacci sequence, nth term formulas, sums, golden ratio limits and how to use this calculator effectively.

This calculator uses the standard convention \(F_0 = 0\) and \(F_1 = 1\). All other terms are generated from the recurrence \(F_n = F_{n-1} + F_{n-2}\) for \(n \ge 2\). Some textbooks instead start with \(F_1 = 1\), \(F_2 = 1\); that convention is equivalent up to a shift of the index, but here the zero-based version is used consistently for all computations and formulas.

The Nth Fibonacci and sum tabs use a fast doubling algorithm together with high-precision integers, so they can handle quite large values of \(n\). For usability and performance in a browser, the input is capped at a maximum index (for example 50 000 for sums and Nth term), which already produces numbers with thousands of digits. Within these limits, the results are exact integers rather than rounded approximations.

There is a classical number-theoretic result that a non-negative integer \(N\) belongs to the Fibonacci sequence if and only if at least one of \(5N^2 + 4\) or \(5N^2 - 4\) is a perfect square. The test tab computes these two expressions using high-precision integers, checks whether either is a perfect square and classifies the input accordingly. This avoids generating all Fibonacci numbers up to \(N\) or searching linearly through the sequence.

The exact values are computed using integer arithmetic and the recurrence, so they are precise. The closed-form expression uses real arithmetic with square roots and powers, so rounding at the floating-point level can introduce small errors. For moderate values of \(n\), these errors are typically less than one-half in magnitude, so rounding the closed-form value to the nearest integer recovers the exact Fibonacci number. The Binet mode compares the two and reports the numerical error explicitly.

Any sequence that satisfies \(a_n = a_{n-1} + a_{n-2}\) with some initial values \(a_0 = A\) and \(a_1 = B\) can be modelled using the custom recurrence tab. This includes the Fibonacci sequence itself (when \(A = 0\), \(B = 1\)) and many related sequences used in population models, financial growth approximations and abstract algebra examples. The calculator computes \(a_n\) and displays a small table of terms so you can inspect how the sequence evolves.

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